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The role of space dimension in reversible charge transfer kinetics in solids
Journal of Luminescence 82 (1999) 129}136
The role of space dimension in reversible charge transfer kinetics in solids M.S. Mikhelashvili* Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel Received 27 August 1998; received in revised form 28 January 1999; accepted 18 March 1999
Abstract The theory of the reversible charge transfer (RCT) for in"nite Euclidean three-dimensional space has been extended to include regular d-dimensional solids, where d"1,2,3. The space dimension determines the character of charge transfer kinetics. The back charge transfer's e!ect depends on the space averaging procedure. Comparison of two space averaging procedures shows the di!erent kinetic behavior in the RCT process in three-dimensional Euclidean space. This di!erence is diminished with the decrease of the space dimension. We explain this fact by increase of the donor's excited state probability N(t) in low-dimensional space. This function distorts the behavior of the charge transfer kinetics in three-dimensional space. The charge separation increased with space dimension's growth. 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Charge transfer probability; Space's dimensionality; Exchange interaction; Excitation power; Charge recombination; Rate constant; Restricted geometry
1. Introduction The search of systems and physicochemical processes, which permit realization of e!ective photoseparation of charges, is a signi"cant problem today in the domain of photocatalitic conversion of solar energy into chemical energy [1}3]. Recently there has been considerable interest and attention focused on excitation and reversible charge transfer (RCT) kinetics for regular and disordered systems [4}13]. The comparison [7,8] of the theories [7}10] and [11] gives approximately the same results under certain conditions. The in#uence of the three-dimensional Euclidean space restriction and
* Fax: #972-2-5618033.
space fractal properties in the RCT kinetics for solids and in the nonlinear luminescence quenching process in the charge transfer phenomena have been discussed in Refs. [12}15]. The in#uence of these aspects for relaxation processes is essential. These conditions are formally analogous to the changing of the Euclidean space dimension d [16}22] and must produce similar e!ects. With each of these factors the space dimension, obviously, plays an important role in the charge transfer and separation processes. The present work is dedicated to this problem for solids. Ionization of the excited donor proceeds according to the kinetic scheme [23]: SH#AP[S>2A\], where A is an electron acceptor. The ions are either recombined or separated. Ionization of the excited
0022-2313/99/$ } see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 0 2 5 - 3
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M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
donors SH and recombination of the ions occur with distant dependent rates k (r) and k (r) due to the exchange interaction [24}26]. When the multipolar interaction is weak, the exchange integral may become signi"cant. The role of space dimensionality is, obviously, essential in RCT kinetics, as its diminution must increase the recombination e!ect; the cation state probability is diminished due to the space restriction [12,13]. Therefore, the study of dimension in#uence is interesting in understanding charge transfer kinetics. This aspect is combined with the space restriction and its regularities in#uence the charge transfer process [4}6,12,13]. Thus, it is interesting and important to study: (a) the in#uence of the space dimension at low excitation power of donors ensemble as well as at high excitation power [15] and comparison of the results of these two conditions of charge separation; (b) the role of an averaging procedure in the RCT process in the spaces of di!erent dimensions. The space dimension's e!ects have been studied in Refs. [16}22] for a #uorescence quenching process. The quencher concentration is very low so that only probe-quencher pairs should be considered. It is a very good approximation when a single excited state is surrounded by not too many noninteracting quenchers, especially in many-dimensional solid spaces. A detailed description of the excitation transfer from the donor to the acceptor into domains of di!erent dimensions has been given in the papers of Blumen et al. [20}22]. The results are given by the d - dimensionality of the lattice and < } the volume B of a unit sphere in d-dimensions for isotropic multipolar interactions and isotropic exchange interactions for solids. For d"1, 2 and 3 we have < "2, B 2p and 4p, respectively. The theory of non-Markovian reversible reactions on one and three dimensions is provided in Refs. [17}19].
2. The reversible charge transfer in three-dimensional solids The probability of the cation state of donor S may be obtained [11}15] using the method "rst proposed by Antonov-Romanovskii and Galanin
[27,28] for electronic energy transfer in solid solutions and subsequently used [29}31] for liquids. At a su$ciently low concentration of acceptors they may be considered as point particles. In this case the di!erential encounter theory describes well the quenching kinetics and quantum yield of luminescence [9]. Let 4pC>(r, t)r dr represents the probability that acceptor anion A\ is located at a distance r, r#dr from the donor molecule S> which is in the cation state [11}15,27}31]. The variation of C>(r, t) with time occurs in solids due to the forward electron transfer between S* and A molecules and the back charge transfer from A\ to S>. The acceptor's concentration C (t) can in the common case also be the function of time. The donor's molecule can exist in three states: excited, ground and cation states. For corresponding probabilities we have N(t)#N (t)#P>(t)"1.
(1)
We are interested in the time dependence of the probability of the donor's non-cation state, N (t) [11}13], as P>(0)"0. Using the method considered in Refs. [11}15,26}31] we introduce for discussion the space averaged probability of noncation state N (t)"N(t)#N (t),
(2)
for which it is easy to write the kinetic equations describing RCT. The probability of the cation state is equal to P>(t)"1!N (t)
(3)
with initial condition P>(0)"0, N (0)"1.
(4)
As the sum of probabilities of donor's excited and ground states is changed due to the forward and back charge transfer, the natural equation for N (t) is
dN (t)/dt"!F(t)4p
*C>(r, t)/*t r dr, 0
*C>(r, t)/*t"k (r) n(r, t)!k (r) C>(r, t),
(5) (6)
M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
where the rate constants of forward and back charge transfer k (r) and k (r) determine the chang ing of C>(r, t). In Eq. (5) the function F(t) demands to be de"ned. From the above de"nition of C>(r, t) follows that F(t) is the concentration of the excited donor } acceptor's pairs, as the integral term in Eq. (5) determines the derivation of donor } cation state probability calculated per one initially excited donor}acceptor pair. If the charge back transfer takes place, we can write in common case of any excitation power of the donor's system F(t)"N (t) ) [C (t)#C (t)] (7) as the function of acceptor's concentration. The correctness of such expression for F(t) will be obvious from the following analysis. So, the donor molecules concentration in the non-cation state is derived from the Bernoulli equation for in"nite space for low excitation power, [C (t)(C (t)] dN (t)/dt"![N (t)] 4pC(t) *C>(r, t)/*t r dr. (8) ; 0 In the case of high excitation power, [C (t)
dN (t)/dt"!N (t) 4pC
*C>(r, t)/*t r dr. 0 (9)
This equation naturally follows from Eqs. (5) and (7). C is the initial concentration of SH. Eq. (8) corresponds to a loss of one excited state. The di!erential operator */*t in the integral of Eq. (5) means that the changing of C>(r, t) occurs due to the charge reversible transfer. The donor excited molecules and the donors in the ground state form part of the donor}molecule system, which is changed due to the forward and back charge transfer. The equations for N (t) are re duced to the well-known equations of the theory of electron energy transfer by Forster}Dexter [24}31]. By proceeding the obvious relationship between probabilities of the donor cation and noncation states (Eq. (3)), we can use the obvious equality of their derivatives PQ >(t)"!NQ (t)
(10)
131
and obtain the donor cation-state probability for low excitation power P>(t)"1!exp[!4p C
C>(r, t) r dr]. (11) 0 Accordingly to the theory [7}10] for P>(t) we have the equations P>(t)"4p C
m(r, t) r dr (12) 0 m (r, t)"k (r)n(r, t) N(t)!k (r) m(r, t), (13) which determine the donor}cation state probability P>(t) at low excitation power [7}10]. Here the m(r, t) function coincides with the C>(r, t) function only for N(t) 1, and this fact results in the qualitative distinction between theories [11] and [7}10]. N(t) } the excited state probability of donors } obeys the conventional kinetic equation of di!erential non}Markovian encounter theory (see Refs. [9,10]) and is determined by the acceptor concentration and the forward charge transfer rate constant [24}31]. n(r, t) is the donor excited state probability with a distance r, r#dr from the acceptor
n (r, t)"!k (r) n(r, t), n(r, 0)"1.
(14) (15)
At time t"0, the acceptor molecules are uniformly distributed, and the function m(r, t) satis"es the initial condition at r'R : m(r, 0)"0. (16) R is the sum of the radii of two molecules. The rate constants, for the Marcus &normal region' of ionization and charge recombination [9,10], are well known as &exchange' transfer [24}26]: k (r)"1/q exp [(R !r)/a ] forward transfer,
(17)
k (r)"1/q exp [(R !r)/a ] back transfer. (18) R and R are the Forster}Dexter characteristic distances [24}31]. a and a are determined by the
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M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
wave function's overlap for a neutral molecules donor}acceptor. The function m(r, t) satis"es Eq. (13) with initial condition (16). Only the approximation N(t)&exp(!t/q)
(19)
for our analysis of RCT in Eqs. (12) and (13) gives a physically reasonable behaviour of the donor } cation state probability for di!erent reaction parameters. At low concentration C of molecu les A or at small k (r) the integral term in Eq. (11) has a small value a. The decomposition of Eq. (11) with a low parameter a gives the result [9,10] } Eq. (12).
3. The role of space dimension (d(3) at low excitation power By low excitation power we mean the following situation [14,15]: (a) the distance between the excited donor molecules SH is so large that the rate of energy (or charge) transfer between SH is much less than the rates of all other processes that deactivate SH; (b) the frequency of possible transfer to a given molecule of acceptor A is small by comparison with the rate of recovery of the capacity to accept energy (or charge after recombination) after energy or charge back transfer. It is clear that the luminescence kinetics will be independent of the excitation power under these conditions. If the concentration of acceptors and donors is small enough to exclude their multi-particle interactions, the di!erential encounter theory (see Ref. [9] and references therein) describes well the quenching kinetics and quantum yield of luminescence. For the study of the role of space dimension in RCT kinetics we restrict ourself to the conventional encounter theory [9]. After pulse excitation of the sample the probability of donor's excited state N(t) obeys the kinetic equation NQ (t)"!k (t) C N(t)!N(t)/q, '
(20)
N(t)"exp(!t/q) q(t).
(21)
q is the lifetime of the excitation. In d-dimensional space
k (t)" k (r) n(r, t) dBr '
(22)
is the nonstationary rate constant of binary ionization which proceeds with a position dependent rate of k (r). Here dBr is an elementary volume in d-dimensional space. For d"1, 2 and 3 we have dBr"2dr, 2pr dr and 4pr dr, respectively. Form of Eqs. (17) and (18) assumes that the transfer time is longer than the phonon relaxation time, so that any excess energy is randomized on the time scale of the transfer. We shall consider only the decay due to the excitation (or charge) transfer. We also suppose that the intramolecular, radiational and radiationless decay channels of the donor are to a good approximation independent of the energy transfer processes, and their e!ect may thus be readily incorporated into the "nal results by multiplication with exp(!t/q). In Eqs. (12) and (13) the donor excited state probability for solid space is determined by the space dimension dependent expression [20}22] ln N(t)"!d< po B ; +1!exp[!tk (r)],rB\ dr, (23) where d is the dimensionality of the lattice and < is B the volume of a unite sphere in d-dimensions. We see that an increase of d results in the decrease of N(t). We obtain the following for an isotropic exchange interaction for d-dimension [20}22]:
N(t)"exp[!< p oc\Bg (t/q exp [cd])], (24) B B where the c"1/a and the g(u) function is de"ned through
g (u)"d +1!exp[!u exp(!y)],yB\ dy. B
(25)
A list of the properties of g(u) is given in Ref. [21]. Eqs. (24) and (25) for d"3 were obtained by Inokuti and Hirayama [29]. For arbitrary d the result is given by Blumen and Silbey [21]. Expressions (11)}(18) are necessary for the analysis of cation
M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
state probability without back electron transfer as well as with back transfer [7}11]. Taking N(0)"1 we can consider P>(t) as the total probability of forward electron transfer at time t and p(r, t) as a density of products around a single donor. In binary approximation P>(t) must be linear in acceptor concentration C [9], and therefore in d-dimensional space
P>(t)" p(r, t)dBr"C
m(r, t) dBr.
(26)
For d-dimensional space dBr"d< rB\ dr. B
(27)
For low acceptor concentrations and long times the results of Inokuty and Hirayama [26], generalized to arbitrary dimensions, are retrieved in Refs. [20}22]. We see that the space dimension determines the behaviour of N(t) (24) and the space integral (26). The total number of charges reaches its maximum value, which is the quantum yield of ionization
P(R)"C m (r) dBr
(28)
and is determined by the space dimension. Here m (r)"m(r,R) is their space distribution: m (r)"k (r)
n(r,t) N(t) dt
(29)
and is a function of d, determined by N(t). Obviously, Eqs. (12) and (13) are correct only for su$ciently low acceptor concentration [9,10]. The space-dimension-dependent term is the "rst part of Eq. (13) and the integral in Eq. (23). The theories [7}10] and [11] make it possible to analyze and to compare the in#uence of the space dimension on the charge transfer kinetics for solids.
4. Discussion The numerical calculations here are complete for solids. We use the numerical procedure based on the expanded DCR program [32,33]. From Eq. (25)
133
follows, that the survival probability of excited donor N(t) is the decreasing function of the space dimension d. Thus the distribution of the counterions arround the ions, which are not determined at the beginning [m(r, 0)"0], is the increasing function of the dimension d (without back transfer, [Eq. (13), k (r)"0]). However, if back transfer takes place (k '0), the function P>(t) } the cation state probability } has the maximum, which removes to the long-time directions when the space dimension d decreases [Fig. 1]. Eqs. (12) and (13) for m(r, t) are correct for a low concentration of acceptors C and for the low k (r) } rate of forward transfer, when we can approximately place N(t) exp(!t/q). m(r, t) does not depend on the space dimension for solids in this case, and cation state dimension dependence is determined only by the space integral in Eq. (12). Naturally, in the limit of in"nite dimension the rate of formation of cations is in"nite, the recombination does not play any role and the cation state probability P>(t)P1, even if k (r)'0, (Fig. 1). In Fig. 1 the dependence P>(t) for di!erent dimensions d"1, 2, 3 of solids is shown. We see the maximum of P>(t) for low dimensions d"1 and 2. For d*3 the dependence P>(t) goes on the stationary plato, the height of which is determined by the C and by the charge transfer parameters R , R , a , a . With an increase of a the maximum of P>(t) removes to the beginning of the coordinate system. The recombination e!ect increases with decrease in dimension and this diminishes the P>(t) curves half-widths; the decrease of P> and nar rowing of P (t) curves. The decrease of the space > dimension gives the same consequences as space restriction [4,5,12]. When there is no back transfer, P(t) increases with space dimension d. In any case, if dimension d grows, independently of availability of the charge back transfer, then cation state probability P>(t)P1 (Fig. 1). The model of the space-averaging procedure [7] gives the physically nonlogical results in the cases when the function q(t)O1 in Eqs. (13), (21) and (26). In particular, we have such results for three-dimensional space; in low-dimensional Euclidean space the cation state kinetics has a physically plausible behavior.
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M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
Fig. 1. Kinetics of photoionization P>(t) for di!erent space dimensions: d"3; 2 and 1 for solids. (a) The theory [7], C "0.1 M/L (bottom up; d"1, 2, 3). (b) The theory [7], C "0.5 M/L. (c) The theory [11], C "0.1 M/L. (d) The theory [11], C "0.5 M/L. d"3 } upper line and d"2 lower curve. R "R "10 A; a "a "1 A, R "5 A. Upper limit of integration B"10 A.
Fig. 2. The dependence of P>(t) for di!erent R "R according to theory [7]. a "1.3 A, a "1 A; C "0.1 M/L. R "5 A, upper limit of the integral } B"10 A. (a) Three-dimensional Euclidean space, d"3. The curve 1, R "R "10 A. The curve 2, R "R "15 A. The curves 3 and 4; R "R "15 A and 10 A, respectively. (b) The same as in (a) according to theory [11]. R "R "0 (lower curves) and 5 A (upper curves). Lower pair curves } d"2 and upper pair of curves } d"3.
M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
135
Fig. 3. The dependence P>(t) for di!erent a . R "R "13 A; a "1 A; C "0.5 M/L, R "5 A. The theory [7], without accounting q. (a) R "R "13 A. d"3. Upper curve } a "1.5 A.; lower curve } a "1 A. (b) d"2; the curves 1 and 2: a "1 and 1.5 A, respectively. The curves 3 and 4: R "13 A and R "10 A.; a "1 and 1.5 A, respectively. P> (t) increases with increase of a .
In Fig. 2 the dependencies P>(R "R ) are shown according to the theory [7] for Euclidean space dimensions d"1, 2, 3. We see the decrease of P> for d"3. However, this curve increases in space with d"1 or 2 when q(t) is approximatly 1. In contrast, according to theory [9] all the curves for d"1, 2, 3 are increasing functions +in the theory [9] for Eqs. (13) and (21) we have: q(t)"1,. The situation is analogous for dependencies P>(a ) for di!erent d (Fig. 3). The recombination e!ect increases with increase of d which diminishes P>(t) curves half-widths. The decrease of P> and
narrowing of P>(t) decreases coming to the stationary plateau. The decrease of the space dimension leads to the same e!ects as the space restriction or its disorder (fractal properties) [4}6]. In three-dimensional space the increase of a results in the increase of the P> (t), while for d"2 [when
q(t)&1, in Eq. (21)] this behavior is opposite and has a plausible character. The coincidence of the results of two di!erent theories [7}10] and [11] under the correct conditions shows that the use of the non-cation state probability N (t) in the theory [11}15] is correct. 5. Conclusions (1) The dimension of the space determines the charge separation kinetics, the e$ciency of which
increases with an increase of the dimension of space. (2) The correctness of the space averaging procedure in the reversible charge transfer theory is elucidated by the space's dimension. (3) The comparison of two di!erent theories shows that in three-dimensional space theory [7] gives incorrect kinetic dependencies for the donor}cation state probability. They have a plausible behavior in the theory [11]. (4) The decreasing recombination branch of probability P>(t) is absent in of the low-dimensional solid space. (5) In all cases, when q(t)+1 and N(t)+ exp(!t/q), both theories [7] and [11] give approximately the same results.
Acknowledgements We wish to thank Profs. J. Klafter for attention and helpful discussions and the Israel Academy of Humanity and Sciences for "nancial support.
References [1] V.M. Agranovich, M.D. Galanin, Electronic Excitation Energy Transfer in Condensed Matter, North-Holland, Amsterdam, 1982.
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[2] G.I. Kavarnos, N.J. Turro, J. Chem. Rev. 86 (1986) 401. [3] K.I. Zamaraev, V.N. Parmon, Usp. Chim. 52 (1983) 1433 (in Russian). [4] J. Klafter, J. Jortner, A. Blumen (Eds.), Dynamical Processes in Condensed Molecular Systems, World Scienti"c, Singapore, 1989. [5] J. Klafter, I.M. Drake (Eds.), Molecular Dynamics in Restricted Geometries, Wiley, New York, 1989. [6] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [7] Y. Lin, R.C. Dorfman, M.D. Fayer, J. Phys. Chem. 90 (1990) 159. [8] R.C. Dorfman, M. Tachiya, M.D. Fayer, Chem. Phys. Lett. 179 (1991) 152. [9] A.I. Burshtein, E.B. Krissinel, M.S. Mikhelashvili, J. Phys. Chem. 98 (1994) 7319. [10] A.I. Burshtein, Chem. Phys. Lett. 194 (1992) 247. [11] M.S. Mikhelashvili, J. Feitelson, M. Dodu, Chem. Phys. Lett. 171 (1990) 575. [12] M.S. Mikhelashvili, Chem. Phys. Lett. 224 (1994) 459. [13] M.S. Mikhelashvili, A.M. Michaely, J. Phys. Chem. 98 (1994) 8114. [14] M.S. Mikhelashvili, E.I. Kapinus, Isr. J. Chem. 33 (1993) 193. [15] S.F. Kilin, M.S. Mikhelashvili, I.M. Rozman, Isv. Akad. Nauk S.S.S.R. Seriya Fiz. 42 (1978) 414. [16] A. Szabo, J. Phys. Chem. 93 (1989) 6929.
[17] N. Agmon, A. Szabo, J. Chem. Phys. 92 (1990) 5270. [18] M. Almgren, J. Alsins, Israel J. Chem. 31 (1991) 159. [19] M. Almgren, J. Alsins, E. Mukhtar, J. van Stam, J. Phys. Chem. 92 (1988) 4479. [20] A. Blumen, IL Nuovo Cimento 63 B (1981) 50. [21] A. Blumen, R. Silbey, J. Chem. Phys. 70 (1979) 3707. [22] A. Blumen, J. Chem. Phys. 72 (1980) 2632. [23] D. Rehm, A. Weller, Isr. J. Chem. 8 (1970) 259. [24] Th. Forster, Z. Na, turforshung. 4a (1949) 311. [25] D.L. Dexter, J. Chem. Phys. 21 (1953) 836. [26] M. Inokuti, F. Hirayama, J. Chem. Phys. 43 (1965) 1978. [27] V.V. Antonov-Romanovski, M.D. Galanin, Opt. I Spectroskopiya 3 (1957) 389. [28] M.D. Galanin, Zh. Eksp. Theoret. Fiz. 28 (1955) 485. [29] S.F. Kilin, M.S. Mikhelashvili, I.M. Rozman, Opt. Spectrosc. (USSR) 16 (1964) 576. [30] M.M. Agrest, S.F. Kilin, M.M. Rikenglas, I.M. Rozman, Opt. Spectrosc. 27 (1969) 946. [31] I.Z. Steinberg, E. Katchalsky, J. Chem. Phys. 48 (1968) 2404. [32] E.B. Krissinel, N.V. Shokhirev, Di!erential approximation of spin-controlled and anisotropic di!usional kinetics, Sibirian Academy Scienti"c Council &Mathematical Methods in Chemistry', 1989, Preprint N 30 (Russian). [33] E.B. Krissinel, N.V. Shokhirev, Di!usion-Controlled Reactions, 1990, Krissinel' and Shokhirev; Inc.; DCR User's Manual, 11-2011990.
The role of space dimension in reversible charge transfer kinetics in solids M.S. Mikhelashvili* Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel Received 27 August 1998; received in revised form 28 January 1999; accepted 18 March 1999
Abstract The theory of the reversible charge transfer (RCT) for in"nite Euclidean three-dimensional space has been extended to include regular d-dimensional solids, where d"1,2,3. The space dimension determines the character of charge transfer kinetics. The back charge transfer's e!ect depends on the space averaging procedure. Comparison of two space averaging procedures shows the di!erent kinetic behavior in the RCT process in three-dimensional Euclidean space. This di!erence is diminished with the decrease of the space dimension. We explain this fact by increase of the donor's excited state probability N(t) in low-dimensional space. This function distorts the behavior of the charge transfer kinetics in three-dimensional space. The charge separation increased with space dimension's growth. 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Charge transfer probability; Space's dimensionality; Exchange interaction; Excitation power; Charge recombination; Rate constant; Restricted geometry
1. Introduction The search of systems and physicochemical processes, which permit realization of e!ective photoseparation of charges, is a signi"cant problem today in the domain of photocatalitic conversion of solar energy into chemical energy [1}3]. Recently there has been considerable interest and attention focused on excitation and reversible charge transfer (RCT) kinetics for regular and disordered systems [4}13]. The comparison [7,8] of the theories [7}10] and [11] gives approximately the same results under certain conditions. The in#uence of the three-dimensional Euclidean space restriction and
* Fax: #972-2-5618033.
space fractal properties in the RCT kinetics for solids and in the nonlinear luminescence quenching process in the charge transfer phenomena have been discussed in Refs. [12}15]. The in#uence of these aspects for relaxation processes is essential. These conditions are formally analogous to the changing of the Euclidean space dimension d [16}22] and must produce similar e!ects. With each of these factors the space dimension, obviously, plays an important role in the charge transfer and separation processes. The present work is dedicated to this problem for solids. Ionization of the excited donor proceeds according to the kinetic scheme [23]: SH#AP[S>2A\], where A is an electron acceptor. The ions are either recombined or separated. Ionization of the excited
0022-2313/99/$ } see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 0 2 5 - 3
130
M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
donors SH and recombination of the ions occur with distant dependent rates k (r) and k (r) due to the exchange interaction [24}26]. When the multipolar interaction is weak, the exchange integral may become signi"cant. The role of space dimensionality is, obviously, essential in RCT kinetics, as its diminution must increase the recombination e!ect; the cation state probability is diminished due to the space restriction [12,13]. Therefore, the study of dimension in#uence is interesting in understanding charge transfer kinetics. This aspect is combined with the space restriction and its regularities in#uence the charge transfer process [4}6,12,13]. Thus, it is interesting and important to study: (a) the in#uence of the space dimension at low excitation power of donors ensemble as well as at high excitation power [15] and comparison of the results of these two conditions of charge separation; (b) the role of an averaging procedure in the RCT process in the spaces of di!erent dimensions. The space dimension's e!ects have been studied in Refs. [16}22] for a #uorescence quenching process. The quencher concentration is very low so that only probe-quencher pairs should be considered. It is a very good approximation when a single excited state is surrounded by not too many noninteracting quenchers, especially in many-dimensional solid spaces. A detailed description of the excitation transfer from the donor to the acceptor into domains of di!erent dimensions has been given in the papers of Blumen et al. [20}22]. The results are given by the d - dimensionality of the lattice and < } the volume B of a unit sphere in d-dimensions for isotropic multipolar interactions and isotropic exchange interactions for solids. For d"1, 2 and 3 we have < "2, B 2p and 4p, respectively. The theory of non-Markovian reversible reactions on one and three dimensions is provided in Refs. [17}19].
2. The reversible charge transfer in three-dimensional solids The probability of the cation state of donor S may be obtained [11}15] using the method "rst proposed by Antonov-Romanovskii and Galanin
[27,28] for electronic energy transfer in solid solutions and subsequently used [29}31] for liquids. At a su$ciently low concentration of acceptors they may be considered as point particles. In this case the di!erential encounter theory describes well the quenching kinetics and quantum yield of luminescence [9]. Let 4pC>(r, t)r dr represents the probability that acceptor anion A\ is located at a distance r, r#dr from the donor molecule S> which is in the cation state [11}15,27}31]. The variation of C>(r, t) with time occurs in solids due to the forward electron transfer between S* and A molecules and the back charge transfer from A\ to S>. The acceptor's concentration C (t) can in the common case also be the function of time. The donor's molecule can exist in three states: excited, ground and cation states. For corresponding probabilities we have N(t)#N (t)#P>(t)"1.
(1)
We are interested in the time dependence of the probability of the donor's non-cation state, N (t) [11}13], as P>(0)"0. Using the method considered in Refs. [11}15,26}31] we introduce for discussion the space averaged probability of noncation state N (t)"N(t)#N (t),
(2)
for which it is easy to write the kinetic equations describing RCT. The probability of the cation state is equal to P>(t)"1!N (t)
(3)
with initial condition P>(0)"0, N (0)"1.
(4)
As the sum of probabilities of donor's excited and ground states is changed due to the forward and back charge transfer, the natural equation for N (t) is
dN (t)/dt"!F(t)4p
*C>(r, t)/*t r dr, 0
*C>(r, t)/*t"k (r) n(r, t)!k (r) C>(r, t),
(5) (6)
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where the rate constants of forward and back charge transfer k (r) and k (r) determine the chang ing of C>(r, t). In Eq. (5) the function F(t) demands to be de"ned. From the above de"nition of C>(r, t) follows that F(t) is the concentration of the excited donor } acceptor's pairs, as the integral term in Eq. (5) determines the derivation of donor } cation state probability calculated per one initially excited donor}acceptor pair. If the charge back transfer takes place, we can write in common case of any excitation power of the donor's system F(t)"N (t) ) [C (t)#C (t)] (7) as the function of acceptor's concentration. The correctness of such expression for F(t) will be obvious from the following analysis. So, the donor molecules concentration in the non-cation state is derived from the Bernoulli equation for in"nite space for low excitation power, [C (t)(C (t)] dN (t)/dt"![N (t)] 4pC(t) *C>(r, t)/*t r dr. (8) ; 0 In the case of high excitation power, [C (t)
dN (t)/dt"!N (t) 4pC
*C>(r, t)/*t r dr. 0 (9)
This equation naturally follows from Eqs. (5) and (7). C is the initial concentration of SH. Eq. (8) corresponds to a loss of one excited state. The di!erential operator */*t in the integral of Eq. (5) means that the changing of C>(r, t) occurs due to the charge reversible transfer. The donor excited molecules and the donors in the ground state form part of the donor}molecule system, which is changed due to the forward and back charge transfer. The equations for N (t) are re duced to the well-known equations of the theory of electron energy transfer by Forster}Dexter [24}31]. By proceeding the obvious relationship between probabilities of the donor cation and noncation states (Eq. (3)), we can use the obvious equality of their derivatives PQ >(t)"!NQ (t)
(10)
131
and obtain the donor cation-state probability for low excitation power P>(t)"1!exp[!4p C
C>(r, t) r dr]. (11) 0 Accordingly to the theory [7}10] for P>(t) we have the equations P>(t)"4p C
m(r, t) r dr (12) 0 m (r, t)"k (r)n(r, t) N(t)!k (r) m(r, t), (13) which determine the donor}cation state probability P>(t) at low excitation power [7}10]. Here the m(r, t) function coincides with the C>(r, t) function only for N(t) 1, and this fact results in the qualitative distinction between theories [11] and [7}10]. N(t) } the excited state probability of donors } obeys the conventional kinetic equation of di!erential non}Markovian encounter theory (see Refs. [9,10]) and is determined by the acceptor concentration and the forward charge transfer rate constant [24}31]. n(r, t) is the donor excited state probability with a distance r, r#dr from the acceptor
n (r, t)"!k (r) n(r, t), n(r, 0)"1.
(14) (15)
At time t"0, the acceptor molecules are uniformly distributed, and the function m(r, t) satis"es the initial condition at r'R : m(r, 0)"0. (16) R is the sum of the radii of two molecules. The rate constants, for the Marcus &normal region' of ionization and charge recombination [9,10], are well known as &exchange' transfer [24}26]: k (r)"1/q exp [(R !r)/a ] forward transfer,
(17)
k (r)"1/q exp [(R !r)/a ] back transfer. (18) R and R are the Forster}Dexter characteristic distances [24}31]. a and a are determined by the
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wave function's overlap for a neutral molecules donor}acceptor. The function m(r, t) satis"es Eq. (13) with initial condition (16). Only the approximation N(t)&exp(!t/q)
(19)
for our analysis of RCT in Eqs. (12) and (13) gives a physically reasonable behaviour of the donor } cation state probability for di!erent reaction parameters. At low concentration C of molecu les A or at small k (r) the integral term in Eq. (11) has a small value a. The decomposition of Eq. (11) with a low parameter a gives the result [9,10] } Eq. (12).
3. The role of space dimension (d(3) at low excitation power By low excitation power we mean the following situation [14,15]: (a) the distance between the excited donor molecules SH is so large that the rate of energy (or charge) transfer between SH is much less than the rates of all other processes that deactivate SH; (b) the frequency of possible transfer to a given molecule of acceptor A is small by comparison with the rate of recovery of the capacity to accept energy (or charge after recombination) after energy or charge back transfer. It is clear that the luminescence kinetics will be independent of the excitation power under these conditions. If the concentration of acceptors and donors is small enough to exclude their multi-particle interactions, the di!erential encounter theory (see Ref. [9] and references therein) describes well the quenching kinetics and quantum yield of luminescence. For the study of the role of space dimension in RCT kinetics we restrict ourself to the conventional encounter theory [9]. After pulse excitation of the sample the probability of donor's excited state N(t) obeys the kinetic equation NQ (t)"!k (t) C N(t)!N(t)/q, '
(20)
N(t)"exp(!t/q) q(t).
(21)
q is the lifetime of the excitation. In d-dimensional space
k (t)" k (r) n(r, t) dBr '
(22)
is the nonstationary rate constant of binary ionization which proceeds with a position dependent rate of k (r). Here dBr is an elementary volume in d-dimensional space. For d"1, 2 and 3 we have dBr"2dr, 2pr dr and 4pr dr, respectively. Form of Eqs. (17) and (18) assumes that the transfer time is longer than the phonon relaxation time, so that any excess energy is randomized on the time scale of the transfer. We shall consider only the decay due to the excitation (or charge) transfer. We also suppose that the intramolecular, radiational and radiationless decay channels of the donor are to a good approximation independent of the energy transfer processes, and their e!ect may thus be readily incorporated into the "nal results by multiplication with exp(!t/q). In Eqs. (12) and (13) the donor excited state probability for solid space is determined by the space dimension dependent expression [20}22] ln N(t)"!d< po B ; +1!exp[!tk (r)],rB\ dr, (23) where d is the dimensionality of the lattice and < is B the volume of a unite sphere in d-dimensions. We see that an increase of d results in the decrease of N(t). We obtain the following for an isotropic exchange interaction for d-dimension [20}22]:
N(t)"exp[!< p oc\Bg (t/q exp [cd])], (24) B B where the c"1/a and the g(u) function is de"ned through
g (u)"d +1!exp[!u exp(!y)],yB\ dy. B
(25)
A list of the properties of g(u) is given in Ref. [21]. Eqs. (24) and (25) for d"3 were obtained by Inokuti and Hirayama [29]. For arbitrary d the result is given by Blumen and Silbey [21]. Expressions (11)}(18) are necessary for the analysis of cation
M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
state probability without back electron transfer as well as with back transfer [7}11]. Taking N(0)"1 we can consider P>(t) as the total probability of forward electron transfer at time t and p(r, t) as a density of products around a single donor. In binary approximation P>(t) must be linear in acceptor concentration C [9], and therefore in d-dimensional space
P>(t)" p(r, t)dBr"C
m(r, t) dBr.
(26)
For d-dimensional space dBr"d< rB\ dr. B
(27)
For low acceptor concentrations and long times the results of Inokuty and Hirayama [26], generalized to arbitrary dimensions, are retrieved in Refs. [20}22]. We see that the space dimension determines the behaviour of N(t) (24) and the space integral (26). The total number of charges reaches its maximum value, which is the quantum yield of ionization
P(R)"C m (r) dBr
(28)
and is determined by the space dimension. Here m (r)"m(r,R) is their space distribution: m (r)"k (r)
n(r,t) N(t) dt
(29)
and is a function of d, determined by N(t). Obviously, Eqs. (12) and (13) are correct only for su$ciently low acceptor concentration [9,10]. The space-dimension-dependent term is the "rst part of Eq. (13) and the integral in Eq. (23). The theories [7}10] and [11] make it possible to analyze and to compare the in#uence of the space dimension on the charge transfer kinetics for solids.
4. Discussion The numerical calculations here are complete for solids. We use the numerical procedure based on the expanded DCR program [32,33]. From Eq. (25)
133
follows, that the survival probability of excited donor N(t) is the decreasing function of the space dimension d. Thus the distribution of the counterions arround the ions, which are not determined at the beginning [m(r, 0)"0], is the increasing function of the dimension d (without back transfer, [Eq. (13), k (r)"0]). However, if back transfer takes place (k '0), the function P>(t) } the cation state probability } has the maximum, which removes to the long-time directions when the space dimension d decreases [Fig. 1]. Eqs. (12) and (13) for m(r, t) are correct for a low concentration of acceptors C and for the low k (r) } rate of forward transfer, when we can approximately place N(t) exp(!t/q). m(r, t) does not depend on the space dimension for solids in this case, and cation state dimension dependence is determined only by the space integral in Eq. (12). Naturally, in the limit of in"nite dimension the rate of formation of cations is in"nite, the recombination does not play any role and the cation state probability P>(t)P1, even if k (r)'0, (Fig. 1). In Fig. 1 the dependence P>(t) for di!erent dimensions d"1, 2, 3 of solids is shown. We see the maximum of P>(t) for low dimensions d"1 and 2. For d*3 the dependence P>(t) goes on the stationary plato, the height of which is determined by the C and by the charge transfer parameters R , R , a , a . With an increase of a the maximum of P>(t) removes to the beginning of the coordinate system. The recombination e!ect increases with decrease in dimension and this diminishes the P>(t) curves half-widths; the decrease of P> and nar rowing of P (t) curves. The decrease of the space > dimension gives the same consequences as space restriction [4,5,12]. When there is no back transfer, P(t) increases with space dimension d. In any case, if dimension d grows, independently of availability of the charge back transfer, then cation state probability P>(t)P1 (Fig. 1). The model of the space-averaging procedure [7] gives the physically nonlogical results in the cases when the function q(t)O1 in Eqs. (13), (21) and (26). In particular, we have such results for three-dimensional space; in low-dimensional Euclidean space the cation state kinetics has a physically plausible behavior.
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M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
Fig. 1. Kinetics of photoionization P>(t) for di!erent space dimensions: d"3; 2 and 1 for solids. (a) The theory [7], C "0.1 M/L (bottom up; d"1, 2, 3). (b) The theory [7], C "0.5 M/L. (c) The theory [11], C "0.1 M/L. (d) The theory [11], C "0.5 M/L. d"3 } upper line and d"2 lower curve. R "R "10 A; a "a "1 A, R "5 A. Upper limit of integration B"10 A.
Fig. 2. The dependence of P>(t) for di!erent R "R according to theory [7]. a "1.3 A, a "1 A; C "0.1 M/L. R "5 A, upper limit of the integral } B"10 A. (a) Three-dimensional Euclidean space, d"3. The curve 1, R "R "10 A. The curve 2, R "R "15 A. The curves 3 and 4; R "R "15 A and 10 A, respectively. (b) The same as in (a) according to theory [11]. R "R "0 (lower curves) and 5 A (upper curves). Lower pair curves } d"2 and upper pair of curves } d"3.
M.S. Mikhelashvili / Journal of Luminescence 82 (1999) 129}136
135
Fig. 3. The dependence P>(t) for di!erent a . R "R "13 A; a "1 A; C "0.5 M/L, R "5 A. The theory [7], without accounting q. (a) R "R "13 A. d"3. Upper curve } a "1.5 A.; lower curve } a "1 A. (b) d"2; the curves 1 and 2: a "1 and 1.5 A, respectively. The curves 3 and 4: R "13 A and R "10 A.; a "1 and 1.5 A, respectively. P> (t) increases with increase of a .
In Fig. 2 the dependencies P>(R "R ) are shown according to the theory [7] for Euclidean space dimensions d"1, 2, 3. We see the decrease of P> for d"3. However, this curve increases in space with d"1 or 2 when q(t) is approximatly 1. In contrast, according to theory [9] all the curves for d"1, 2, 3 are increasing functions +in the theory [9] for Eqs. (13) and (21) we have: q(t)"1,. The situation is analogous for dependencies P>(a ) for di!erent d (Fig. 3). The recombination e!ect increases with increase of d which diminishes P>(t) curves half-widths. The decrease of P> and
narrowing of P>(t) decreases coming to the stationary plateau. The decrease of the space dimension leads to the same e!ects as the space restriction or its disorder (fractal properties) [4}6]. In three-dimensional space the increase of a results in the increase of the P> (t), while for d"2 [when
q(t)&1, in Eq. (21)] this behavior is opposite and has a plausible character. The coincidence of the results of two di!erent theories [7}10] and [11] under the correct conditions shows that the use of the non-cation state probability N (t) in the theory [11}15] is correct. 5. Conclusions (1) The dimension of the space determines the charge separation kinetics, the e$ciency of which
increases with an increase of the dimension of space. (2) The correctness of the space averaging procedure in the reversible charge transfer theory is elucidated by the space's dimension. (3) The comparison of two di!erent theories shows that in three-dimensional space theory [7] gives incorrect kinetic dependencies for the donor}cation state probability. They have a plausible behavior in the theory [11]. (4) The decreasing recombination branch of probability P>(t) is absent in of the low-dimensional solid space. (5) In all cases, when q(t)+1 and N(t)+ exp(!t/q), both theories [7] and [11] give approximately the same results.
Acknowledgements We wish to thank Profs. J. Klafter for attention and helpful discussions and the Israel Academy of Humanity and Sciences for "nancial support.
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